Partially-blind Estimation of Reciprocal Channels for AF Two-Way Relay Networks Employing M-PSK Modulation

We consider the problem of channel estimation for amplify-and-forward two-way relays assuming channel reciprocity and M-PSK modulation. In an earlier work, a partially-blind maximum-likelihood estimator was derived by treating the data as deterministic unknowns. We prove that this estimator approaches the true channel with high probability at high signal-to-noise ratio (SNR) but is not consistent. We then propose an alternative estimator which is consistent and has similarly favorable high SNR performance. We also derive the Cramer-Rao bound on the variance of unbiased estimators.

💡 Research Summary

The paper addresses channel estimation in amplify‑and‑forward (AF) two‑way relay (TWR) networks under the assumptions of channel reciprocity and M‑ary phase‑shift keying (M‑PSK) modulation. In such networks two terminals exchange information via a single relay that simply amplifies its received signal and forwards it. Because the relay does not decode, the channels from each terminal to the relay (h₁ and h₂) are multiplied by the same amplification factor and, under reciprocity, h₂ is the complex conjugate of h₁. Accurate knowledge of these channels is essential for coherent detection and for optimal power allocation, yet the relay’s blind amplification makes estimation challenging.

Prior work and its limitation
The authors first revisit a previously proposed “partially‑blind maximum‑likelihood (ML) estimator.” In that approach the transmitted data symbols are treated as deterministic but unknown parameters, and the log‑likelihood of the received complex samples is maximized jointly over the channel and the symbols. By exploiting the structure of the M‑PSK constellation, a closed‑form expression for the ML estimate of the channel can be derived. The paper proves that, at high signal‑to‑noise ratio (SNR), the estimator converges to the true channel with high probability. However, the authors demonstrate that the estimator is inconsistent: even as the number of observations N → ∞, the estimate does not converge in probability to the true channel. The inconsistency stems from the deterministic‑unknown treatment of the data symbols, which introduces a structural bias that does not vanish with more samples. Consequently, in practical systems where long observation windows are used to average out noise, the ML estimator would still retain a non‑zero error floor.

Proposed consistent estimator
To overcome this drawback, the paper proposes a new “consistent partially‑blind estimator.” The key idea is to model the transmitted symbols as random variables with a known prior (uniform over the M‑PSK constellation) rather than as fixed unknowns. Using Bayes’ rule, the posterior probability of each symbol given the received sample and a tentative channel value is computed. The expectation of the symbol under this posterior, ( \mathbb{E}